In this paper, written for non specialists, we discuss several points in the elementary theory of almost complex manifolds, with a focus on the question of choice of special coordinates and on the obstruction given by the Nijenhuis tensor.

If X is an almost complex manifold, with an almost complex structure J of class C a, for some a>0, then for every point pEX and every tangent vector V at p, there exists a germ of J-holomorphic discs through p with this prescribed tangent vector. This existence result goes back to [NW] The example is very explicit and very simple to describe. See Part II. We refer the reader unfamiliar with the above notions to [IR]. For the proof of the failure of upper semi-continuity we shall need the following result.(1) Partly supported by an NSF grant.

We will denote by U (resp. Ū) the open (resp. closed) unit disk in C. Chirka [1] (see also [2]) recently proved the following remarkable result.

Abstract. This paper is an outgrowth of a paper by the first author on a generalized Hartogs Lemma. We complete the discussion of the nonlinear ∂ problem ∂f /∂z = ψ(z, f (z)). We also simplify the proofs by a different choice of Banach spaces of functions.

J -Holomorphic curves are −∞ sets of J -plurisubharmonic functions, with a singularity of LogLog type, but it is shown that in general they are not −∞ sets of J -plurisubharmonic functions with Logarithmic singularity (i.e. non-zero Lelong number). Some few additional remarks on pluripolarity in almost complex structures are made. Mathematics Subject Classification (2000)32Q60 · 32Q65 · 32U05 0 Introduction J -Plurisubharmonic functions with poles (at which the function is −∞) have already played a role in almost complex analysis. They has been used in the study of the Kobayashi metric [4,5] for getting an efficient control of J -holomorphic discs. In a work in progress with Ivashkovich, applications to uniqueness problems are given. The first pluripolarity result is due to Chirka who showed that if J is a C 1 almost complex structure defined near 0 in C n and if J (0) = J st (the standard complex structure), then for A > 0 large enough log |z| + A|z| is J -plurisubharmonic near 0. A complete proof has been written in [5, Lemma 1.4, page 2400]. The function − log | log |z|| is also J -plurisubharmonic near 0. Although a 'log-log singularity' (zero Lelong number!) is much less interesting and has less applications that a 'log singularity', functions with log-log singularity were introduced in [10] to show pluripolarity of J -holomorphic curves. Later, Elkhadhra [2] generalized the result to show pluri-polarity of J -holomorphic submanifolds, again with a 'loglog' singularity of the function.In [10] there has been an error in stating the smoothness hypotheses, all smoothness requirements in the statements have to be increased by 1 (C k+1 instead of C k ). Indeed on line 11-page 663 it is claimed that because J [Y , J Y ](Z , 0) = 0, one has |J [Y , J Y ](Z , Z )| ≤ C|Z ||Y | 2 . This is correct if J is of class C 2 (or at least C 1,1 ), but C 1 smoothness, hence continuity of [Y , J Y ], is not enough.